Classes of maps with continuous selections and homotopy extension

This paper presents a variety of homotopy principles for multivalued maps with continuous selections based on null–homotopic, extendability, essentiality and the Urysohn function. In particular we first present two fixed point results for a multimap ϕ where (i). ϕ has a continuous single–valued selection f, (ii). f is homotopic to a constant map, and (iii). a ”null–homotopic compact self map” property holds. Next some examples of the null–homotopic property are given via the Arens-Eells theorem and the Schauder fixed point theorem. The final part of the paper discusses essential maps in a very general setting and in this setting we establish that if two multivalued maps (with continuous selections) are homotopic and one is an essential map then the other map has a fixed point.